472 ' Mr, wales 3/2 the Rcfolutkn 
Thus, let 22 be io, and 3 24 ; and the difference of the loga- 
rithms of a and \b will be 0.0791812. Now, by running the 
eye along ga^diner’s Tables of logarithmic Sines and Tan- 
py. 
its, hr will be readily feen, that the log:, tangent of 26 
1 j 00 
° f // 
33 5 ° 
when me) 
o / 
ifed by 0.079 1S1 2, 
A2 
j J 
4o v twice that arc 
is lefs than the cofine of 
and that the k>p\ tangent of 
^ U 
6° 34' cD, when increaled by the fame quantity, is too great, 
And, by actually taking out the logarithms, and making the 
additions, the former will be found too fmail by 45;, and the 
latter too great by 632. Then, 1087 (455 + 632) : 10'' :: 
455 : 4 -" 5 which being added to 26° 33' 50'' gives 26° 33' 54" 
for the arc of which x is the co-tangent. And if to twice the 
log. co- tangent of this arc the logarithm of a (10) he added, 
the fum (1.6020600) is the log. of 40, the value of a fought. 
EXAMPLE VI. 
The equation refulting from a folution of the famous problem 
of ALHA2EN may be given as another example of the life of this 
method. Many folution s of this celebrated problem, by huy- 
gens, slustus, and others, may be met with in the Philofo- 
phical Tnmfadiions. Solutions to it may alfo be found at ther 
end of Dr. RonRRT simpson’s Conic Sections, in Dr. smith’s 
O ptics, Mr. robin’s . Mathematical Trafts, and other places ; 
but the moft direct and obvious method is, perhaps, that which 
follows. 
Put a == DC; b = t/C, r = CI ; x = CB andy = CE, the cofines 
of the arcs I A,. 1 H, to the radius r: then will the fines of 
thofe arcs,. BA, EH, be expreffed bv %/' r z - x 2 and V r~ — ) 2 ; 
and, 
